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Number of results
Journal
2009 | 7 | 3 | 591-600
Article title

Labyrinthine pathways towards supercycle attractors in unimodal maps

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EN
Abstracts
EN
As an important preceding step for the demonstration of an uncharacteristic (q-deformed) statisticalmechanical structure in the dynamics of the Feigenbaum attractor we uncover previously unknown properties of the family of periodic superstable cycles in unimodal maps. Amongst the main novel properties are the following: i) The basins of attraction for the phases of the cycles develop fractal boundaries of increasing complexity as the period-doubling structure advances towards the transition to chaos. ii) The fractal boundaries, formed by the pre-images of the repellor, display hierarchical structures organized according to exponential clusterings that manifest in the dynamics as sensitivity to the final state and transient chaos. iii) There is a functional composition renormalization group (RG) fixed-point map associated with the family of supercycles. iv) This map is given in closed form by the same kind of q-exponential function found for both the pitchfork and tangent bifurcation attractors. v) There is final-stage ultra-fast dynamics towards the attractor, with a sensitivity to initial conditions which decreases as an exponential of an exponential of time. We discuss the relevance of these properties to the comprehension of the discrete scale-invariance features, and to the identification of the statistical-mechanical framework present at the period-doubling transition to chaos. This is the first of three studies (the other two are quoted in the text) which together lead to a definite conclusion about the applicability of q-statistics to the dynamics associated to the Feigenbaum attractor.
Publisher
Journal
Year
Volume
7
Issue
3
Pages
591-600
Physical description
Dates
published
1 - 9 - 2009
online
25 - 6 - 2009
References
  • [1] H. G. Schuster, Deterministic Chaos. An Introduction, 2nd Revised Edition (VCH Publishers, Weinheim, 1988)
  • [2] A. Robledo, Europhysics News 36, 214 (2005) http://dx.doi.org/10.1051/epn:2005611[Crossref]
  • [3] A. Robledo, arXiv:0710.1047 [WoS]
  • [4] P. Grassberger, R. Badii, A. Politi, J. Stat. Phys. 51, 135 (1988) http://dx.doi.org/10.1007/BF01015324[Crossref]
  • [5] H. Mori, H. Hata, T. Horita, T. Kobayashi, Prog. Theor. Phys. Supp. 99, 1 (1989) http://dx.doi.org/10.1143/PTPS.99.1[Crossref]
  • [6] D. J. Evans, G. P. Morris, Statistical Mechanics of Nonequilibrium Liquids, Theoretical Chemistry Monograph Series (Academic Press, London, 1990)
  • [7] C. Beck, F. Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, UK, 1993) http://dx.doi.org/10.1017/CBO9780511524585[Crossref]
  • [8] R. C. Hilborn, Chaos and Nonlinear Dynamics, 2nd Revised Edition (Oxford University Press, New York, 2000) http://dx.doi.org/10.1093/acprof:oso/9780198507239.001.0001[Crossref]
  • [9] A. Robledo, Physica A 314, 437 (2002) http://dx.doi.org/10.1016/S0378-4371(02)01177-9[Crossref]
  • [10] A. Robledo, Physica D 193, 153 (2004) http://dx.doi.org/10.1016/j.physd.2004.01.016[Crossref]
  • [11] F. Baldovin, A. Robledo, Europhys. Lett. 60, 518 (2002) http://dx.doi.org/10.1209/epl/i2002-00249-7[Crossref]
  • [12] B. Hu, J. Rudnick, Phys. Rev. Lett. 48, 1645 (1982) http://dx.doi.org/10.1103/PhysRevLett.48.1645[Crossref]
  • [13] C. Grebogi, S. W. McDonald, E. Ott, J. A. Yorke, Phys. Lett. 99A, 415 (1983)
  • [14] S. W. McDonald, C. Grebogi, E. Ott, J. A. Yorke, Physica D 17, 125 (1985) http://dx.doi.org/10.1016/0167-2789(85)90001-6[Crossref]
  • [15] E. Mayoral, A. Robledo, Phys. Rev. E 72, 026209 (2005) http://dx.doi.org/10.1103/PhysRevE.72.026209[Crossref]
  • [16] H. Hernández-Saldaña, A. Robledo, Physica A 370, 286 (2006) http://dx.doi.org/10.1016/j.physa.2006.03.018[Crossref]
  • [17] A. Robledo, Physica A 370, 449 (2006) http://dx.doi.org/10.1016/j.physa.2006.06.003[Crossref]
  • [18] A. Robledo, L. G. Moyano, arXiv:0706.4422 [WoS]
  • [19] D. Sornette, Phys. Rep. 297, 239 (1998) http://dx.doi.org/10.1016/S0370-1573(97)00076-8[Crossref]
  • [20] A. Robledo, L. G. Moyano, Phys. Rev. E 77, 032613 (2008) http://dx.doi.org/10.1103/PhysRevE.77.036213[Crossref]
Document Type
Publication order reference
YADDA identifier
bwmeta1.element.-psjd-doi-10_2478_s11534-009-0065-1
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